The extent of a droplet's spreading over a flat, smooth solid substrate and its equilibrium height in the presence of gravity are determined approximately, without a numerical solution of the governing nonlinear differential equation, by assuming that the droplet takes on the shape of an oblate spheroidal cap and by minimizing the corresponding free energy. The comparison with the full numerical evaluations confirms that the introduced approximation and the obtained results are accurate for contact angles below about 120° and for droplet sizes on the order of the capillary length of the liquid. The flattening effect of gravity is to increase the contact radius and decrease the height of the droplet, with these being more pronounced for higher values of the Bond number.

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The shape of a uniformly rotating liquid droplet deposited on a solid substrate is determined by an iterative numerical integration of the governing nonlinear differential equation. The differential equation and the boundary conditions are derived by means of the variational analysis which delivers the expressions for the specific configurational force per unit area of the liquid/vapor interface, and the configurational force along the liquid/solid/vapor contact circle. An analytical proof for the orthogonality of the specific configurational force to the surface of the droplet is constructed. The effect of rotation on the droplet’s gyrostatic shape is discussed

Skid-steered, tracked, tele-operated robots are used to perform high-risk critical missions such as bomb disposal under conditions deemed too risky to send a human. Often the robots carry heavy payloads that raise their centers of mass, increasing the risk of tip-over. Since it is often not feasible to send a human to right a toppled robot, tipover is equivalent to mission failure. Hence, an autonomous behavior to prevent robot tip-over is desired. In this research, a simplified model of mobile robot dynamics permits separation of pitch and roll stabilization. Adaptive control is used to stabilize the appropriate angle based on the normalized tip-over measure. Experimental validation of this control is successfully demonstrated on an iRobot Packbot and a Segway RMP 440.

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